1. Technical Field
This invention relates to the art of modeling manufacturing systems, and more particularly to the art of developing analytical models for stochastic manufacturing systems and using such models to optimize process variables within the system.
2. Discussion of the Prior Art
In serial manufacturing systems, workpieces or goods flow through stages that are separated by storage spaces for temporary storage, referred to herein as buffers. Each stage comprises one or more machining operations, a robotically-generated task, or a task performed by human operation such as assembly or machining. Buffers can be either parallel or crossover. The crossover type is capable of storing and cross-feeding workpieces from any upstream stage to any downstream stage, and the parallel type can only store and feed workpieces from a single designated upstream stage to a designated downstream stage. Since each stage has its own cycle time, frequency of machine breakdown, and time required to repair, and each buffer has its own capacity, the flow of the system can be interrupted, starved, or blocked by any mismatches between stages.
The problem is to maximize the performance of such system, but its performance is governed by a combination of interrelated variables. Optimizing one variable without considering its interrelationship to the other variables leads to little or no improvement in performance. Thus, the maximization solution lies in finding the combination of variables under which the apparatus of the system meets a desired performance. Some of these variables are, of course, more controllable than others. So the degree of design freedom may be limited and constraints may exist even on the controllable variables. For example, with fixed cycle times, the design may involve determining the number of machines needed for each operation and, perhaps, subject to cost and space constraints. On the other hand, for a given set of breakdown/repair characteristics, optimization of the buffer size is of primary concern.
The first part of the problem is thoroughly understanding the behavior of the system by creating an accurate stochastic model that reflects the randomness of certain of the variables. The second part of the problem is to integrate the model with a method of rapidly determining values for process variables that meet a desired optimization objective. With respect to the first part of the problem, we must set apart prior art not related to continuous manufacturing systems as being irrelevant (such as in U.S. Pat. Nos. 4,710,864 and 4,604,718).
One of the earliest stochastic modeling approaches uses discrete event simulation which is based on logical descriptions of the system, devoid of math (see U.S. Pat. No. 4,796,194). While the discrete event simulation approach is capable of accurately modeling manufacturing systems, it does not lend itself to optimization. It is strictly an evaluation tool. Given a set of parameters, a simulation is run and the system's behavior and performance predicted. However, there is no mechanism, beyond trial and error based on the intuition of the user, for estimating how the parameters should be changed in order to achieve the desired objective.
Analytical models of manufacturing systems are attractive because they may offer "closed form solutions" which lend themselves to use with established optimization procedures. The term "analytical" refers to the fact that the system behavior is described by precise mathematical equations. Queuing models are included in this category (see the following articles: Solberg, J. J., "A Mathematical Model of Computerized Manufacturing Systems", Proceedings of 4th International Conference on Production Research, Tokyo, Japan, August, 1977; Stecke, K. E., "Production Planning Problems for Flexible Manufacturing Systems", PHD Thesis, Purdue University, West Lafayette, Ind., 1981. An analytical model based on decomposition and approximation was set forth in an article by Gershwin, S. B., "An Efficient Method for the Approximate Evaluation of Production Lines With Finite Storage Space", Proceedings of 23rd IEEE Conference on Decision and Control, Las Vegas, Nev., December, 1984. This latter model is limited by the number of machines and type of service distribution that it can handle and therefore is inadequate.
Finally, still another analytical modeling approach was set forth in Buzacott, J. A., "Automatic Transfer Lines with Buffer Stocks", International Journal of Production Research, 5, 3, pp. 183-200, 1967. The difficulty with these previous approaches is that they all place restrictions on the system behavior which limits their application to actual manufacturing systems. For example, queuing theory usually assumes a fixed arrival pattern of work to a server, while decomposition methods apply only to systems with a very small number of machines. Buzacott's work assumed infinite buffers and thus are inappropriate for optimizing buffer size.
With respect to the second part of the problem, the prior art is possessed of a variety of optimization procedures among which is included a simplex method, gradient search procedures, and trajectory methods (see U.S. Pat. No. 4,744,027). These methods were all developed for deterministic problems and can be set apart as inapplicable to stochastic problems. Deterministic problems are those which, for the same set of variable values, yield the same answer. Stochastic systems, however, have random components. In the absence of infinitely long simulation times (which are impractical), each run of the model can contain a different sequence of random events and give rise to different answers. In such systems, the objective function estimate becomes very "noisy" and parameter optimization quite difficult. It is difficult to ascertain how much of an observed change is due to the parameter change and how much is due to random "noise". Under such conditions, the aforementioned optimization methods perform poorly. The direction of the search is often mislead and convergence is poor, if it is even achieved at all.